m = 0.31ρ
-1.91
(Eq 2)
where is expressed as a fraction of the density of pore-free material. The value of m for pore-free pure iron is 0.31, and
any excess over this value for porous iron is due to geometric work hardening.
A further consequence of the densification of porous preforms during deformation is reflected in Poisson's ratio for the
porous material. Poisson's ratio is a measure of the lateral flow behavior of a material; for compression of a cylinder, it is
expressed as diametral strain
d
divided by height strain -
z
. For a pore-free material, Poisson's ratio for plastic
deformation is 0.5. This is a direct result of the fact that the volume of the material remains constant during
deformation. For example, equating the volume of a cylinder before and after deformation (Ref 44):
H
0
[ )/4] = H
f
[( )/4]
(Eq 3)
where H
0
and H
f
are initial and final cylinder heights, respectively, and D
0
and D
f
are initial and final cylinder diameters,
respectively. Dividing by H
f
yields:
H
0
/H
f
= (D
f
/D
0
)
2
(Eq 4)
and taking logarithms
ln (H
0
/H
f
) = ln (D
f
/D
0
)
2
= 2 ln (D
f
/D
0
)
(Eq 5)
or
-
z
= 2
d
(Eq 6)
and, from the definition of Poisson's ratio:
(Eq 7)
During compressive deformation of a sintered metal powder preform, some material flows into the pores, and there is a
volume decrease. For a given reduction in height, the diameter of a sintered P/M cylinder will expand less than that of an
identical cylinder of a pore-free material. Poisson's ratio for a P/M preform will therefore be less than 0.5 and will be a
function of the pore volume fraction. H.A. Kuhn (Ref 45) has established an empirical relationship between Poisson's
ratio and part density:
= 0.5
a
(Eq 8)
The best fit to experimental data is obtained with the exponent a = 1.92 for room-temperature deformation and a = 2.0 for
hot deformation. The slight difference in this exponent may be due to work hardening (Ref 44).
In deformation processing of materials, plasticity theory is useful for calculating forming pressures and stress
distributions. The above mentioned idiosyncrasies in the deformation behavior of sintered, porous materials have been
taken into account in the development of a plasticity theory for porous materials. This has been of benefit in applying
workability analysis to porous preforms (Ref 31, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59).
A typical workability line is shown in Fig. 11, which also indicates the way processing variables affect the location of the
line. The line has a slope of -0.5, and workability improves as the plane strain intercept (y-axis intercept) value increases.
For a given material, workability can be improved through either temperature adjustment or a change in preform density.
Figure 12(b) to (e) show the effects of temperature and pressure on the densification and forming of the powder forged